### 数学代写|随机过程统计代写Stochastic process statistics代考|STAT3021

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|随机过程统计代写Stochastic process statistics代考|Constraint Method

In the constraint method, the optimization problem represented in Equation $1.9$ is modified as follows:
optimize $Z_{i}$
s.t.
\begin{aligned} &Z_{j} \leq \varepsilon_{j} \quad(j=1,2, \ldots, k ; j \neq i) \ &h(\bar{x})=0 \ &g(\bar{x}) \leq 0 \end{aligned}
Thus, one of the objective functions is selected to be optimized, and the others are moved to the set of constraints as inequality constraints, with a right-side term $\varepsilon_{j}$. As a first step, individual optimization problems are solved for the objectives $Z_{j}$, obtaining the lower and upper limits for each of those objectives. Then, using the range of values for the individual objectives, the problem is discretized and a given number of internal points are selected. The selection of the data can be performed through a sampling methodology. Each point will be a set of values for the $\varepsilon_{j}$ terms. Thus, for $n$ selected points, $n$ optimization problems should be solved, and each solution will be a nondominated one.

For a more detailed discussion of the presented multiobjective optimization methods, the reader is referred to the study of Diwekar (2010). Other multiobjective optimization methods are consulted in the study of Marler and Arora (2004).

## 数学代写|随机过程统计代写Stochastic process statistics代考|Process Optimization

In general, engineering systems are typically good candidates for optimization, owing to its high number of degrees of freedom. Furthermore, the equations modeling such systems are typically nonlinear and may involve both algebraic and differential relationships. Thus, a rigorous approach to solve optimization problems in engineering is mandatory. Chemical engineering, and in particular, process engineering is not an exception. Imagining the heart of a chemical process, the reactor, such systems are typically modeled by nonlinear equations, including, as an example, the Arrhenius equation to represent the changes on the kinetic constant with temperature. Separation systems are also represented by models with a high number of nonlinear equations, e.g., the thermodynamic relationships modeling phase equilibrium. Furthermore, when the unit operations consist of various separation stages, the number of equations is increased. The number of variables in the models of chemical processes and the number of degrees of freedom are high. Variables such as temperature, pressure, mass of catalyst, holdup, coolant flow rate, among others are typically degrees of freedom in process engineering. In the case of the objective functions, there are some typical objectives in process engineering, such as minimizing the total annual costs, maximizing the profit, minimizing the environmental impact, maximizing the social impact, and minimizing the control effort. Constraints are typically provided by the mathematical models itself, but also by the inherent characteristics of the process, e.g., lower and upper limits for the variables. Lower limits are typically provided by the positive nature of most of the physical variables; whereas, upper limits are provided by operational limitations. When designing equipment, constraints may also occur owing to limitations on available space for installation. From the last lines, it can be deduced that process engineering is a source of various optimization problems, where most of them are multivariable ones. Thus, robust strategies are required for solving such situations, considering the high number of degrees of freedom that can be implied. In general terms, process optimization problems can be solved by using three approaches: mathematical programming (Grossmann et al., 1999; Caballero and Grossmann, 2004), stochastic optimization methods (Androulakis and Venkatasubramanian, 1991; Cardoso et al., 2000), and hybrid methods (Banga et al., 2003; Mohammadhasani Khorasany and Fesanghary, 2009). The first approach is the most rigorous approach, since it solves the entire model and finds the optimum solution through the Calculus principles. In the second approach, the solution is searched in the entire feasible region, using some criteria to reach the global optimum. In the third approach, a stochastic method is initially used to reach a zone close to the global optimum, and then a mathematical programming method is used to ensure reaching the global optimum.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Single-Variable Deterministic Optimization

We start with the simplest type of mathematic optimization, which is an unconstrained, one-variable optimization problem. This is helpful to better understand the solution methods of multivariable problems, with and without constraints. The well-known first derivative criterion is that, if we have a given function $\mathrm{f}(x)$, we can find a stationary point $x^{}$ if we derive the function and equal the derivative to zero, and then solve for $x$. That stationary point could be an optimal (minimum or maximum) or not, depending on the characteristics of the function. If the function is convex for any $x$ (the second derivative is positive or zero), then $x^{}$ is a minimum. If the function is concave for any $x$ (the second derivative is negative), then $x^{*}$ is a minimum.

Nevertheless, if the function is neither concave nor convex, then the stationary point is neither a maximum nor a minimum, but a saddle point. Some examples are discussed in this chapter before dealing with multivariable optimization.

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## MATLAB代写

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